André–Quillen cohomology

In commutative algebra, André–Quillen cohomology is a theory of cohomology for commutative rings which is closely related to the cotangent complex. The first three cohomology groups were introduced by Lichtenbaum & Schlessinger (1967) and are sometimes called Lichtenbaum–Schlessinger functors T⁰, T¹, T², and the higher groups were defined independently by Michel André and by Daniel Quillen using methods of homotopy theory. It comes with a parallel homology theory called André–Quillen homology.

Motivation

Let A be a commutative ring, B be an A-algebra, and M be a B-module. André–Quillen cohomology is the derived functors of the derivation functor Der_A(B, M). Before the general definitions of André and Quillen, it was known for a long time that given morphisms of commutative rings A → B → C and a C-module M, there is a three-term exact sequence of derivation modules:

0 \to \operatorname{Der}_B(C, M) \to \operatorname{Der}_A(C, M) \to \operatorname{Der}_A(B, M).

This term can be extended to a six-term exact sequence using the functor Exalcomm of extensions of commutative algebras and a nine-term exact sequence using the Lichtenbaum–Schlessinger functors. André–Quillen cohomology extends this exact sequence even further. In the zeroth degree, it is the module of derivations; in the first degree, it is Exalcomm; and in the second degree, it is the second degree Lichtenbaum–Schlessinger functor.

Definition

Let B be an A-algebra, and let M be a B-module. Let P be a simplicial cofibrant A-algebra resolution of B. André notates the qth cohomology group of B over A with coefficients in M by H^q(A, B, M), while Quillen notates the same group as D^q(B/A, M). The qth André–Quillen cohomology group is:

D^q(B/A, M) = H^q(A, B, M) \stackrel{\text{def}}{=} H^q(\operatorname{Der}_A(P, M)).

The qth André–Quillen homology group is:

D_q(B/A, M) = H_q(A, B, M) \stackrel{\text{def}}{=} H_q(\Omega_{P/A} \otimes_B M).

Let L_B/A denote the relative cotangent complex of B over A. Then we have the formulas:

D^q(B/A, M) = H^q(\operatorname{Hom}_B(L_{B/A}, M)),

D_q(B/A, M) = H_q(L_{B/A} \otimes_B M).

References

André, Michel (1974), Homologie des Algèbres Commutatives, Grundlehren der mathematischen Wissenschaften 206, Springer-Verlag
Lichtenbaum, Stephen; Schlessinger, M. (1967), "The cotangent complex of a morphism", Transactions of the American Mathematical Society 128: 41–70, doi:10.2307/1994516, ISSN 0002-9947, MR 0209339
Quillen, Daniel G., Homology of commutative rings, unpublished notes
Quillen, Daniel (1970), On the (co-)homology of commutative rings, Proc. Symp. Pure Mat. XVII, American Mathematical Society
Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, ISBN 978-0-521-43500-0, MR 1269324

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